Problem: Determine how many solutions exist for the system of equations. ${-10x-2y = -10}$ ${y = -1-6x}$
Solution: Convert both equations to slope-intercept form: ${-10x-2y = -10}$ $-10x{+10x} - 2y = -10{+10x}$ $-2y = -10+10x$ $y = 5-5x$ ${y = -5x+5}$ ${y = -1-6x}$ ${y = -6x-1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -5x+5}$ ${y = -6x-1}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.